Contact Residual Dynamics

• Contact Residual Dynamics (CRD) is a residual modeling approach that learns discrepancies between a nominal model and actual contact behavior in robotics.
• It applies learned corrections to angular dynamics, complementarity constraints, and contact impulses, thereby enhancing aspects like thruster-assisted locomotion and tactile sensing.
• CRD is integrated into predictive controllers using structured physical priors and both deterministic and stochastic methods to improve system performance under uncertainty.

Searching arXiv for papers on Contact Residual Dynamics and closely related residual contact modeling. Tool call: arxiv_search(query="Contact Residual Dynamics MPC learned contact residuals thruster locomotion residual contact model", max_results=10, sort_by="relevance") Contact Residual Dynamics (CRD) denotes a class of residual modeling methods for contact-rich robotic systems in which discrepancies between a nominal or prior model and actual contact behavior are learned and then injected back into prediction, estimation, or control. In the cited literature, CRD appears in several technically distinct forms: as learned residual angular dynamics caused by leg-ground impacts in thruster-assisted quadrupedal locomotion (Wang et al., 5 Aug 2025), as an additive residual in contact complementarity constraints for adaptive contact-implicit MPC (Huang et al., 2023), as residual corrections to analytical contact impulses for long-horizon prediction and uncertainty propagation (Fazeli et al., 2020), and, in a broader sensory formulation, as residual tactile representations that encode the discrepancy between visual priors and physical contact sensations (Zhang et al., 3 Jul 2026). A later line of work further situates CRD within stochastic residual modeling by arguing that contact residuals are often multi-modal rather than merely poorly modeled, and therefore benefit from structured stochastic representations (Kotecha et al., 9 Mar 2026).

1. Conceptual scope and definitions

CRD is fundamentally a residual-dynamics formulation for contact phenomena that are difficult to capture with nominal rigid-body models, simplified actuated models, or visually predicted interaction priors. The recurring technical pattern is decomposition: a structured nominal model captures the dominant predictable component, while a learned residual captures unmodeled interaction effects such as impacts, frictional variability, hybrid mode mismatch, or visually unpredictable tactile events (Wang et al., 5 Aug 2025).

In thruster-enhanced locomotion, the residual is defined at the level of body angular acceleration. The true angular acceleration is written as the sum of the nominal effect of thrusters and residual effects due to leg-ground impact (Wang et al., 5 Aug 2025): $$\dot{\boldsymbol{\omega}}_{\text{true}} = \dot{\boldsymbol{\omega}}_{\text{nominal}} + \dot{\boldsymbol{\omega}}_{\text{residual}}.$$ There, CRD specifically approximates impulsive torques from ground contacts that affect the robot’s orientation, especially roll (Wang et al., 5 Aug 2025).

In adaptive contact-implicit MPC, CRD is defined differently. The residual is inserted additively into the contact complementarity constraints of a local Linear Complementarity System (LCS), where the vector $r_{\text{comp}}$ represents model discrepancies or errors in the contact complementarity constraints and is learned online to minimize predictive error (Huang et al., 2023). In that formulation, CRD primarily adapts the hybrid model boundaries, meaning the transitions between contact and no-contact modes (Huang et al., 2023).

In residual contact simulation, the residual is a corrective impulse added to an analytical contact impulse model. Given a nominal analytical impulse $\mathbf{p}_m$, the learned contact residual is $\mathbf{p}_{res}$, yielding

$$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$

with the post-contact state updated by the corrected impulse (Fazeli et al., 2020). This places CRD at the interface between analytical simulators and empirical data.

A broader interpretation appears in contact-rich manipulation with multimodal sensing. There, tactile information is reformulated as a residual quantity,

$r_t = z_t - \hat{z}_t,$

where the residual captures the discrepancy between visually predicted tactile latent state and actual tactile perception (Zhang et al., 3 Jul 2026). This is presented as a residual representation of unexpected contact information rather than a force or state residual in a dynamics equation.

Taken together, these uses indicate that CRD is not a single fixed model class. It is a family of residual formulations centered on contact-induced discrepancy, with the residual placed at different levels of the pipeline: continuous dynamics, hybrid constraints, contact impulses, or cross-modal sensory representation.

2. Mathematical formulations across representative CRD systems

The most explicit dynamics-based CRD formulation in locomotion appears in the Husky-$\beta$ system, where nominal angular acceleration comes from thruster torques (Wang et al., 5 Aug 2025): $$I\,\dot{\boldsymbol{\omega}}_{\text{nominal}} = \sum_{i=1}^4 \mathbf{r}_i \times \left(c_f\, v_i^2\, \mathbf{\hat{e}}_i\right).$$ The residual component approximates the effect of ground reaction forces at the legs,

$$\dot{\boldsymbol{\omega}}_{\text{residual}} \approx I^{-1} \sum_{i=1}^4 \left(\mathbf{d}_i \times \mathbf{F}_i\right),$$

and the learned version becomes

$$\dot{\boldsymbol{\omega}}_{\text{residual}} = I^{-1} \sum_{i=1}^4 C_{i,\text{NN}} \left( \mathbf{d}_i \times \mathbf{F}_{i,\text{NN}} \right),$$

where each leg contributes a predicted ground reaction force weighted by a predicted contact probability (Wang et al., 5 Aug 2025). The resulting augmented thrust model is

$r_{\text{comp}}$0

This formulation preserves a physically structured nominal term and adds a contact residual torque term inside the predicted angular dynamics (Wang et al., 5 Aug 2025).

In adaptive hybrid MPC, the prior multi-contact model is represented as an LCS,

$r_{\text{comp}}$1

and CRD modifies the complementarity relation to

$r_{\text{comp}}$2

In this case, the residual is not primarily a force correction but a correction to the hybrid contact law itself (Huang et al., 2023).

In residual point-contact learning, the analytical model produces

$r_{\text{comp}}$3

and the contact-corrected post-impact velocity is

$r_{\text{comp}}$4

This formulation is localized to contact events and emphasizes residual learning at the impulse level rather than along continuous dynamics (Fazeli et al., 2020).

A later extension argues that deterministic contact residuals are insufficient when contact outcomes are multi-modal. STRIDE decomposes predicted acceleration into a deterministic Lagrangian component and a stochastic residual force sample (Kotecha et al., 9 Mar 2026): $r_{\text{comp}}$5 This work explicitly states that traditional CRD associates unmodeled effects such as contacts and friction with deterministic, state-dependent residuals, and presents stochastic residuals as an extension that avoids averaging bias (Kotecha et al., 9 Mar 2026).

3. Learning architectures and training objectives

The CRD learning architecture in thruster-enhanced locomotion uses one subnetwork per leg, with four identical neural nets in total (Wang et al., 5 Aug 2025). Each subnetwork processes a 21D feature vector comprising joint angles and velocities, foot and propeller positions, base orientation, angular and linear velocities, and thruster actions (Wang et al., 5 Aug 2025). Each network outputs a predicted 3D ground reaction force $r_{\text{comp}}$6 and a predicted contact probability $r_{\text{comp}}$7, and the effective force per leg is $r_{\text{comp}}$8 (Wang et al., 5 Aug 2025).

Its training objective is physics-informed. The force loss is

$r_{\text{comp}}$9

the contact loss is binary cross-entropy,

$\mathbf{p}_m$0

and the overall loss is

$\mathbf{p}_m$1

The model is trained offline with data from simulation and/or real hardware, and when no contact sensors are available on hardware, simulation-trained weights are used with the contact head frozen (Wang et al., 5 Aug 2025).

In adaptive contact-implicit MPC, CRD is learned online rather than offline. The pipeline collects tuples $\mathbf{p}_m$2, recomputes local LCS matrices at each operating point, defines a batch loss measuring mismatch between predictions and observations, and updates $\mathbf{p}_m$3 using gradient descent or Adam (Huang et al., 2023). The loss has a state-prediction term and a complementarity-violation term: $\mathbf{p}_m$4 with each sample loss involving minimization over nonnegative contact force and slack variables (Huang et al., 2023). The paper states that the gradient with respect to $\mathbf{p}_m$5 is computed using implicit differentiation and that residual learning runs at approximately $\mathbf{p}_m$6 Hz while MPC runs at approximately $\mathbf{p}_m$7 Hz (Huang et al., 2023).

The residual point-contact learner adopts a density neural network that maps contact features to a distribution over corrective impulses (Fazeli et al., 2020). The feature vector uses effective inertia at contact and contact point velocity rather than the full state, yielding a 5-dimensional planar feature representation (Fazeli et al., 2020). The model is stochastic, with the residual impulse sampled from a Gaussian whose mean and covariance depend on the input features. Training minimizes the expected trajectory discrepancy between observed and estimated trajectories and uses gradient-free optimization because contact timing is noisy and semi-Markovian (Fazeli et al., 2020).

STRIDE’s training objective differs by jointly learning a structured Lagrangian prior and a stochastic residual process. The Conditional Flow Matching module generates a residual by integrating a conditional ODE vector field from latent Gaussian noise (Kotecha et al., 9 Mar 2026): $\mathbf{p}_m$8 and training minimizes a supervised regression objective on accelerations (Kotecha et al., 9 Mar 2026). The paper explicitly frames this as an extension beyond deterministic CRD.

4. Integration into model predictive control and predictive systems

A central role of CRD is to improve model-plant alignment inside predictive controllers. In the Husky-$\mathbf{p}_m$9 locomotion architecture, control is decoupled because a unified MPC optimizing both ground reaction forces and thruster forces is limited by the low torque-control bandwidth of lightweight actuators (Wang et al., 5 Aug 2025). The legged controller is Raibert-type and position-based, while an MPC regulates the thrusters using a simplified dynamics model augmented with CRD (Wang et al., 5 Aug 2025). Dynamics are linearized and discretized as

$\mathbf{p}_{res}$0

and the CRD output is incorporated into the reference state used over the MPC horizon: $\mathbf{p}_{res}$1 The MPC then solves

$\mathbf{p}_{res}$2

Because $\mathbf{p}_{res}$3 includes CRD-informed roll, pitch, and yaw trajectories, the reference better aligns prediction with actual dynamics during ground-contact events (Wang et al., 5 Aug 2025).

In adaptive contact-implicit MPC, integration is more direct: the MPC always uses the latest updated residual-augmented LCS model $\mathbf{p}_{res}$4 to compute control inputs (Huang et al., 2023). This allows the controller to adapt the hybrid model boundary online, rather than only correcting continuous dynamics (Huang et al., 2023). The framework thereby couples online system identification of contact discrepancies with real-time contact-implicit control.

Outside MPC, CRD also enters long-horizon prediction pipelines. In residual point-contact learning, the residual is applied only when a collision occurs; between contacts, free motion is propagated deterministically (Fazeli et al., 2020). This yields a contact-aware predictive simulator in which uncertainty is injected at contact events and then propagated through state belief updates (Fazeli et al., 2020). A plausible implication is that this event-triggered residual placement can reduce unnecessary correction outside contact phases, which is consistent with the paper’s emphasis on sample efficiency and physically structured corrections (Fazeli et al., 2020).

5. Empirical behavior, performance, and operational significance

In thruster-enhanced locomotion, offline test predictions show that the CRD-augmented model achieves lower RMSE in angular acceleration $\mathbf{p}_{res}$5, especially for roll, than a nominal-only model (Wang et al., 5 Aug 2025). In push-recovery experiments with a $\mathbf{p}_{res}$6N disturbance applied for $\mathbf{p}_{res}$7s along the $\mathbf{p}_{res}$8-axis, the CRD-augmented controller successfully recovers to a stable trot, whereas the nominal-only controller fails to recover stability (Wang et al., 5 Aug 2025). In hardware narrow-path cat gait trials, the CRD-augmented controller achieves lower RMSE in roll, pitch, and yaw than the nominal dynamics alone, with the largest improvements in roll stability (Wang et al., 5 Aug 2025). In standard trot gait, it maintains stable roll with less required thruster force than the nominal-only case, which the paper interprets as better alignment with the true system dynamics (Wang et al., 5 Aug 2025).

In adaptive multi-contact manipulation, the online residual learning framework is reported to adapt on-the-fly with an adaptation rate around $\mathbf{p}_{res}$9 Hz and to run MPC around $$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$0 Hz (Huang et al., 2023). Hardware experiments show that with a rough prior model, the framework successfully manipulates previously unknown objects with non-smooth surface geometries (Huang et al., 2023). The reported qualitative outcome is that model-based control without adaptation fails, whereas the adaptive CRD-enhanced version enables required contact behavior on hardware (Huang et al., 2023).

In long-horizon residual point-contact learning, the proposed point-contact residual model converges to about $$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$1–$$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$2 lower RMSE over full trajectories than the bare simulator, and does so with as few as $$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$3–$$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$4 trajectories (Fazeli et al., 2020). The paper also reports that a generic recurrent residual baseline, SIAN, requires the entire dataset before matching bare simulator performance (Fazeli et al., 2020). The same work emphasizes uncertainty “blossoming” after each contact and uses Gaussian-mixture belief propagation to represent the resulting multi-modality (Fazeli et al., 2020).

STRIDE reports a $$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$5 reduction in long-horizon prediction error and a $$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$6 reduction in contact force prediction error compared to deterministic residual baselines (Kotecha et al., 9 Mar 2026). On the Unitree Go1 benchmark summarized in the paper, STRIDE attains state RMSE $$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$7 and force error $$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$8, compared with $$\mathbf{p}_{opt} = \mathbf{p}_m + \mathbf{p}_{res},$$9 and $r_t = z_t - \hat{z}_t,$0 for DeLaN and $r_t = z_t - \hat{z}_t,$1 and $r_t = z_t - \hat{z}_t,$2 for LNN + Diffusion (Kotecha et al., 9 Mar 2026). The paper attributes these gains to physically structured modeling and stochastic residual representation of contact phenomena (Kotecha et al., 9 Mar 2026).

In tactile manipulation, ResTacVLA reports $r_t = z_t - \hat{z}_t,$3 average success on a diverse task suite, compared with $r_t = z_t - \hat{z}_t,$4 for the best baseline and $r_t = z_t - \hat{z}_t,$5 for a vision-only model (Zhang et al., 3 Jul 2026). The paper further reports that removing vector quantization reduces average performance by $r_t = z_t - \hat{z}_t,$6, and removing surprise-aware gating reduces it by $r_t = z_t - \hat{z}_t,$7 (Zhang et al., 3 Jul 2026). Although this is not a dynamics-equation formulation of CRD, it shows that residual contact representations can also improve robustness to unexpected dynamic disturbances in action models (Zhang et al., 3 Jul 2026).

6. Deterministic versus stochastic CRD, and related interpretations

A significant conceptual development in the literature is the distinction between deterministic residual correction and stochastic residual modeling. In locomotion CRD and online hybrid MPC, the residual is deterministic once conditioned on state, action, and learned parameters (Wang et al., 5 Aug 2025, Huang et al., 2023). In residual point-contact learning, the residual is explicitly stochastic and represented as a Gaussian correction at contact events, with uncertainty propagated through the subsequent state (Fazeli et al., 2020). STRIDE makes this distinction explicit and argues that deterministic residual methods average over incompatible outcomes in multi-modal settings such as slipping versus sticking (Kotecha et al., 9 Mar 2026). The paper calls this limitation “averaging bias” and presents stochastic residuals as a remedy (Kotecha et al., 9 Mar 2026).

This produces two broad interpretations of CRD. The first treats CRD as a deterministic, state-dependent correction to a nominal model. The second treats CRD as an uncertain or multi-modal interaction term whose distribution must itself be modeled. The available papers support both views. A plausible synthesis is that deterministic CRD is adequate when contact discrepancies are repeatable and sufficiently observable, whereas stochastic CRD becomes necessary when unobserved micro-conditions make contact outcomes inherently variable (Kotecha et al., 9 Mar 2026, Fazeli et al., 2020).

Another interpretive distinction concerns where the residual is inserted. Force-level CRD modifies torques, impulses, or generalized forces (Wang et al., 5 Aug 2025, Fazeli et al., 2020, Kotecha et al., 9 Mar 2026). Constraint-level CRD modifies complementarity relations and therefore contact mode transitions (Huang et al., 2023). Representation-level CRD encodes the mismatch between sensory expectation and sensed contact (Zhang et al., 3 Jul 2026). This suggests that CRD is best understood as a residual-contact principle rather than a single canonical equation.

7. Relationship to adjacent research directions and open issues

CRD is closely related to residual physics, data-augmented simulation, contact-implicit control, and multimodal contact representation. The residual point-contact framework explicitly combines an analytical contact simulator with learned corrections, emphasizing that purely analytical models cannot match observed diversity of outcomes while generic data-driven models require substantially more data (Fazeli et al., 2020). Adaptive contact-implicit MPC extends residual learning into online hybrid control, addressing the difficulty of acquiring accurate multi-contact models without extensive offline tuning (Huang et al., 2023). Thruster-enhanced locomotion uses CRD to compensate for the infeasibility of unified torque-level MPC under actuator bandwidth limitations (Wang et al., 5 Aug 2025). STRIDE, in turn, connects CRD to structured mechanics and generative stochastic modeling (Kotecha et al., 9 Mar 2026).

The literature also reveals several recurring technical challenges. One is hardware observability: in Husky-$r_t = z_t - \hat{z}_t,$8, transfer learning is needed when no contact sensors are available on hardware, and the contact head is frozen during transfer (Wang et al., 5 Aug 2025). Another is real-time optimization: adaptive hybrid MPC requires concurrent residual learning and control loops, with residual updates at around $r_t = z_t - \hat{z}_t,$9 Hz and MPC at around $\beta$0 Hz (Huang et al., 2023). A third is contact multi-modality and uncertainty propagation, which motivates Gaussian or flow-based residual models in the simulation and structured stochastic modeling literature (Fazeli et al., 2020, Kotecha et al., 9 Mar 2026).

A common misconception is to equate CRD exclusively with force estimation. The surveyed formulations show that CRD may estimate forces, but it may also correct hybrid constraints, refine analytical impulses, or encode cross-modal contact discrepancy (Huang et al., 2023, Fazeli et al., 2020, Zhang et al., 3 Jul 2026). Another misconception is that residuals merely patch poor models in an ad hoc manner. The strongest examples in this literature retain explicit physical structure: nominal thruster dynamics plus residual torque (Wang et al., 5 Aug 2025), LCS contact laws plus residual complementarity term (Huang et al., 2023), or Lagrangian mechanics plus stochastic residual interactions (Kotecha et al., 9 Mar 2026).

The current evidence suggests that CRD is most effective when it is physically situated: tied to contact geometry, contact timing, or hybrid contact constraints rather than learned as an unrestricted black-box correction. A plausible implication is that future CRD systems will continue to combine structured priors with residual terms that are event-aware, uncertainty-aware, and suitable for direct insertion into planners or controllers. The available papers already point in that direction through physics-informed residual losses (Wang et al., 5 Aug 2025), online residual adaptation inside contact-implicit MPC (Huang et al., 2023), uncertainty-aware contact correction (Fazeli et al., 2020), and stochastic structured residual dynamics (Kotecha et al., 9 Mar 2026).